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A380089 The number of unitary divisors of n that are terms in A207481.

%I A380089 #10 Jan 12 2025 03:52:58
%S A380089 1,2,2,2,2,4,2,1,2,4,2,4,2,4,4,1,2,4,2,4,4,4,2,2,2,4,2,4,2,8,2,1,4,4,
%T A380089 4,4,2,4,4,2,2,8,2,4,4,4,2,2,2,4,4,4,2,4,4,2,4,4,2,8,2,4,4,1,4,8,2,4,
%U A380089 4,8,2,2,2,4,4,4,4,8,2,2,1,4,2,8,4,4,4
%N A380089 The number of unitary divisors of n that are terms in A207481.
%H A380089 Amiram Eldar, <a href="/A380089/b380089.txt">Table of n, a(n) for n = 1..10000</a>
%F A380089 a(n) = A034444(A380088(n)).
%F A380089 Multiplicative with a(p^e) = 2 if e <= p, and 1 otherwise.
%F A380089 a(n) = 1 if and only if n is in A054743.
%F A380089 a(n) < A034444(n) if and only if n is in A185359.
%F A380089 a(n) = A034444(n) if and only if n is in A207481.
%F A380089 a(n) = A377519(n) if and only if n is squarefree (A005117).
%F A380089 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((p+1)*s)).
%F A380089 Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(p+1))) = 0.93168306734008028353...
%t A380089 f[p_, e_] := If[e <= p, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A380089 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~,if(f[i,2] <= f[i,1], 2, 1));}
%Y A380089 Cf. A005117, A054743, A077610, A185359, A207481, A377519, A380086, A380088, A380090.
%K A380089 nonn,easy,mult
%O A380089 1,2
%A A380089 _Amiram Eldar_, Jan 12 2025